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Theorem opelopabsb 4467
Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
opelopabsb  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Distinct variable groups:    x, y    x, B
Allowed substitution hints:    ph( x, y)    A( x, y)    B( y)

Proof of Theorem opelopabsb
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2961 . . . . . . . . . 10  |-  x  e. 
_V
2 vex 2961 . . . . . . . . . 10  |-  y  e. 
_V
31, 2opnzi 4435 . . . . . . . . 9  |-  <. x ,  y >.  =/=  (/)
4 simpl 445 . . . . . . . . . . 11  |-  ( (
(/)  =  <. x ,  y >.  /\  ph )  ->  (/)  =  <. x ,  y >. )
54eqcomd 2443 . . . . . . . . . 10  |-  ( (
(/)  =  <. x ,  y >.  /\  ph )  ->  <. x ,  y
>.  =  (/) )
65necon3ai 2646 . . . . . . . . 9  |-  ( <.
x ,  y >.  =/=  (/)  ->  -.  ( (/)  =  <. x ,  y
>.  /\  ph ) )
73, 6ax-mp 8 . . . . . . . 8  |-  -.  ( (/)  =  <. x ,  y
>.  /\  ph )
87nex 1565 . . . . . . 7  |-  -.  E. y ( (/)  =  <. x ,  y >.  /\  ph )
98nex 1565 . . . . . 6  |-  -.  E. x E. y ( (/)  =  <. x ,  y
>.  /\  ph )
10 elopab 4464 . . . . . 6  |-  ( (/)  e.  { <. x ,  y
>.  |  ph }  <->  E. x E. y ( (/)  =  <. x ,  y >.  /\  ph ) )
119, 10mtbir 292 . . . . 5  |-  -.  (/)  e.  { <. x ,  y >.  |  ph }
12 eleq1 2498 . . . . 5  |-  ( <. A ,  B >.  =  (/)  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  (/)  e.  { <. x ,  y >.  |  ph } ) )
1311, 12mtbiri 296 . . . 4  |-  ( <. A ,  B >.  =  (/)  ->  -.  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
1413necon2ai 2651 . . 3  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  <. A ,  B >.  =/=  (/) )
15 opnz 4434 . . 3  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
1614, 15sylib 190 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  ( A  e.  _V  /\  B  e.  _V )
)
17 sbcex 3172 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  A  e.  _V )
18 spesbc 3244 . . . 4  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  E. x [. B  / 
y ]. ph )
19 sbcex 3172 . . . . 5  |-  ( [. B  /  y ]. ph  ->  B  e.  _V )
2019exlimiv 1645 . . . 4  |-  ( E. x [. B  / 
y ]. ph  ->  B  e.  _V )
2118, 20syl 16 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  B  e.  _V )
2217, 21jca 520 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  ( A  e.  _V  /\  B  e.  _V )
)
23 opeq1 3986 . . . . 5  |-  ( z  =  A  ->  <. z ,  w >.  =  <. A ,  w >. )
2423eleq1d 2504 . . . 4  |-  ( z  =  A  ->  ( <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. A ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
25 dfsbcq2 3166 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] [ w  /  y ] ph  <->  [. A  /  x ]. [ w  /  y ] ph ) )
2624, 25bibi12d 314 . . 3  |-  ( z  =  A  ->  (
( <. z ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ z  /  x ] [ w  /  y ] ph )  <->  ( <. A ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ]. [ w  / 
y ] ph )
) )
27 opeq2 3987 . . . . 5  |-  ( w  =  B  ->  <. A ,  w >.  =  <. A ,  B >. )
2827eleq1d 2504 . . . 4  |-  ( w  =  B  ->  ( <. A ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } ) )
29 dfsbcq2 3166 . . . . 5  |-  ( w  =  B  ->  ( [ w  /  y ] ph  <->  [. B  /  y ]. ph ) )
3029sbcbidv 3217 . . . 4  |-  ( w  =  B  ->  ( [. A  /  x ]. [ w  /  y ] ph  <->  [. A  /  x ]. [. B  /  y ]. ph ) )
3128, 30bibi12d 314 . . 3  |-  ( w  =  B  ->  (
( <. A ,  w >.  e.  { <. x ,  y >.  |  ph } 
<-> 
[. A  /  x ]. [ w  /  y ] ph )  <->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph ) ) )
32 nfopab1 4276 . . . . . 6  |-  F/_ x { <. x ,  y
>.  |  ph }
3332nfel2 2586 . . . . 5  |-  F/ x <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }
34 nfs1v 2184 . . . . 5  |-  F/ x [ z  /  x ] [ w  /  y ] ph
3533, 34nfbi 1857 . . . 4  |-  F/ x
( <. z ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ z  /  x ] [ w  /  y ] ph )
36 opeq1 3986 . . . . . 6  |-  ( x  =  z  ->  <. x ,  w >.  =  <. z ,  w >. )
3736eleq1d 2504 . . . . 5  |-  ( x  =  z  ->  ( <. x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. z ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
38 sbequ12 1945 . . . . 5  |-  ( x  =  z  ->  ( [ w  /  y ] ph  <->  [ z  /  x ] [ w  /  y ] ph ) )
3937, 38bibi12d 314 . . . 4  |-  ( x  =  z  ->  (
( <. x ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ w  /  y ] ph )  <->  ( <. z ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ z  /  x ] [ w  /  y ] ph ) ) )
40 nfopab2 4277 . . . . . . 7  |-  F/_ y { <. x ,  y
>.  |  ph }
4140nfel2 2586 . . . . . 6  |-  F/ y
<. x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }
42 nfs1v 2184 . . . . . 6  |-  F/ y [ w  /  y ] ph
4341, 42nfbi 1857 . . . . 5  |-  F/ y ( <. x ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ w  /  y ] ph )
44 opeq2 3987 . . . . . . 7  |-  ( y  =  w  ->  <. x ,  y >.  =  <. x ,  w >. )
4544eleq1d 2504 . . . . . 6  |-  ( y  =  w  ->  ( <. x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  <. x ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
46 sbequ12 1945 . . . . . 6  |-  ( y  =  w  ->  ( ph 
<->  [ w  /  y ] ph ) )
4745, 46bibi12d 314 . . . . 5  |-  ( y  =  w  ->  (
( <. x ,  y
>.  e.  { <. x ,  y >.  |  ph } 
<-> 
ph )  <->  ( <. x ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ w  /  y ] ph ) ) )
48 opabid 4463 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
4943, 47, 48chvar 1969 . . . 4  |-  ( <.
x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] ph )
5035, 39, 49chvar 1969 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ z  /  x ] [ w  /  y ] ph )
5126, 31, 50vtocl2g 3017 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<-> 
[. A  /  x ]. [. B  /  y ]. ph ) )
5216, 22, 51pm5.21nii 344 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653   [wsb 1659    e. wcel 1726    =/= wne 2601   _Vcvv 2958   [.wsbc 3163   (/)c0 3630   <.cop 3819   {copab 4267
This theorem is referenced by:  brabsb  4468  opelopabaf  4480  opelopabf  4481  difopab  5008  isarep1  5534  opelopabgf  28073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269
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